#7927401

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#36266	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4290 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,5,4)(3,7,8,6)(9,10)(11,12)(13,21)(14,18)(15,16)(17,23)(19,24)(20,22)(25,26)(27,38)(28,35)(29,40)(30,33)(31,36)(32,37)(34,39);(1,3,5,8)(2,6,4,7)(9,11,15,14)(10,12,16,18)(13,19,23,22)(17,20,21,24)(25,27)(26,31)(28,29)(30,39)(32,36)(33,40)(34,35)(37,38);(1,4,5,2)(3,7,8,6)(10,16)(12,18)(17,21)(20,24)(25,28,32,30)(26,33,37,35)(27,39,36,29)(31,34,38,40);(9,12,15,18)(10,11,16,14)(13,20,23,24)(17,19,21,22)(25,29,32,39)(26,34,37,40)(27,28,36,30)(31,35,38,33);(9,13,15,23)(10,17,16,21)(11,19,14,22)(12,20,18,24)(25,30,32,28)(26,35,37,33)(27,39,36,29)(31,34,38,40);(9,14,15,11)(10,18,16,12)(13,22,23,19)(17,24,21,20)(25,31,32,38)(26,36,37,27)(28,34,30,40)(29,35,39,33)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=g,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7834597);